The Complexity of Finding Tangles
Abstract
We study the following combinatorial problem. Given a set of y-monotone curves, which we call wires, a tangle determines the order of the wires on a number of horizontal layers such that any two consecutive layers differ only in swaps of neighboring wires. Given a multiset of swaps (that is, unordered pairs of wires) and an initial order of the wires, a tangle realizes if each pair of wires changes its order exactly as many times as specified by . Deciding whether a given multiset of swaps admits a realizing tangle is known to be NP-hard [Yamanaka et al., CCCG 2018]. We prove that this problem remains NP-hard if every pair of wires swaps only a constant number of times. On the positive side, we improve the runtime of a previous exponential-time algorithm. We also show that the problem is in NP and fixed-parameter tractable with respect to the number of wires.
Cite
@article{arxiv.2002.12251,
title = {The Complexity of Finding Tangles},
author = {Oksana Firman and Philipp Kindermann and Boris Klemz and Alexander Ravsky and Alexander Wolff and Johannes Zink},
journal= {arXiv preprint arXiv:2002.12251},
year = {2024}
}
Comments
This paper has been superseded by arXiv:2312.16213 (merged from arXiv:1901.06548 and this paper). This paper has appeared in Proceedings of the 48th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2023): https://doi.org/10.1007/978-3-031-23101-8_1